Intersections of powers of a principal ideal and primality

AbstractWe say that an integral domain R satisfies property (∗) if the ideal ⋂n>0anR is prime, for every non-unit a∈R. We investigate property (∗) in the classical situation when R is the integral closure of a valuation domain V in a finite extension L of the field of fractions Q of V. Let f be the irreducible polynomial of an integral element x such that L=Q[x]. Assuming that the discriminant of f is a unit, we prove that R is not a valuation domain if f has roots modulo P, the maximal ideal of V. Then we show that R does not satisfy (∗) if f has roots in V modulo J, for a suitable non-maximal prime ideal J≠0 of V. Moreover, if f has degree 2 or 3 the converses of the above results are true. Examples show that these converses are no longer valid for any degree n⩾4